2,833 research outputs found
Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?
In the literature, necessary and sufficient conditions in terms of
variational inequalities are introduced to characterize minimizers of convex
set valued functions with values in a conlinear space. Similar results are
proved for a weaker concept of minimizers and weaker variational inequalities.
The implications are proved using scalarization techniques that eventually
provide original problems, not fully equivalent to the set-valued counterparts.
Therefore, we try, in the course of this note, to close the network among the
various notions proposed. More specifically, we prove that a minimizer is
always a weak minimizer, and a solution to the stronger variational inequality
always also a solution to the weak variational inequality of the same type. As
a special case we obtain a complete characterization of efficiency and weak
efficiency in vector optimization by set-valued variational inequalities and
their scalarizations. Indeed this might eventually prove the usefulness of the
set-optimization approach to renew the study of vector optimization
Variational inequalities characterizing weak minimality in set optimization
We introduce the notion of weak minimizer in set optimization. Necessary and
sufficient conditions in terms of scalarized variational inequalities of
Stampacchia and Minty type, respectively, are proved. As an application, we
obtain necessary and sufficient optimality conditions for weak efficiency of
vector optimization in infinite dimensional spaces. A Minty variational
principle in this framework is proved as a corollary of our main result.Comment: Includes an appendix summarizing results which are submitted but not
published at this poin
A Minty variational principle for set optimization
Extremal problems are studied involving an objective function with values in
(order) complete lattices of sets generated by so called set relations.
Contrary to the popular paradigm in vector optimization, the solution concept
for such problems, introduced by F. Heyde and A. L\"ohne, comprises the
attainment of the infimum as well as a minimality property. The main result is
a Minty type variational inequality for set optimization problems which
provides a sufficient optimality condition under lower semicontinuity
assumptions and a necessary condition under appropriate generalized convexity
assumptions. The variational inequality is based on a new Dini directional
derivative for set-valued functions which is defined in terms of a "lattice
difference quotient": A residual operation in a lattice of sets replaces the
inverse addition in linear spaces. Relationships to families of scalar problems
are pointed out and used for proofs: The appearance of improper scalarizations
poses a major difficulty which is dealt with by extending known scalar results
such as Diewert's theorem to improper functions
Two approaches toward constrained vector optimization and identity of the solutions
In this paper we deal with a Fritz John type constrained vector optimization problem. In spite that there are many concepts of solutions for an unconstrained vector optimization problem, we show the possibility “to double” the number of concepts when a constrained problem is considered. In particular we introduce sense I and sense II isolated minimizers, properly efficient points, efficient points and weakly efficient points. As a motivation leading to these concepts we give some results concerning optimality conditions in constrained vector optimization and stability properties of isolated minimizers and properly efficient points. Our main investigation and results concern relations between sense I and sense II concepts. These relations are proved mostly under convexity type conditions. Key words: Constrained vector optimization, Optimality conditions, Stability, Type of solutions and their identity, Vector optimization and convexity type conditions.
Increase-along-rays property for vector functions
In this paper we extend to the vector case the notion of increasing along rays function. The proposed definition is given by means of a nonlinear scalarization through the so-called oriented distance function from a point to a set. We prove that the considered class of functions enjoys properties similar to those holding in the scalar case, with regard to optimization problems, relations with (generalized) convex functions and characterization in terms of Minty type variational inequalities. Key words: generalized convexity, increase-along-rays property, star-shaped set, Minty variational inequality.
Variational inequalities in vector optimization
In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called ”oriented distance” function [14, 15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis was needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4].
Minty variational inequalities, increase-along-rays property and optimization
Let E be a linear space, K E and f : K ? R. We put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ,K), which can be considered as a generalization of the Minty variational inequality of differential type (for short, MV I(f ,K)). We investigate, in the case of K star-shaped (for short, st-sh), the existence of a solution x of GMV I(f ,K) and the property of f to increase-along-rays starting at x (for short, f IAR(K, x )). We prove that GMV I(f ,K) with radially l.s.c. function f has a solution x ker K if and only if f IAR(K, x ). Further, we prove, that the solution set of GMV I(f ,K) is a convex and radially closed subset of kerK. We show also that, if GMV I(f ,K) has a solution x K, then x is a global minimizer of the problem f(x) ? min, x K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove, that in case of a quasi-convex function f, these sets coincide. Key words: Minty variational inequality, Generalized variational inequality, Existence of solutions, Increase along rays, Quasi-convex functions.
First order optimality conditions in set-valued optimization
A a set-valued optimization problem minC F(x), x 2 X0, is considered, where X0 X, X and Y are Banach spaces, F : X0 Y is a set-valued function and C Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x0, y0), y0 2 F(x0), and are called minimizers. In particular the notions of w-minimizer (weakly efficient points), p-minimizer (properly efficient points) and i-minimizer (isolated minimizers) are introduced and their characterization in terms of the so called oriented distance is given. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive first order conditions, that is conditions in terms of suitable first order derivatives of F, for a pair (x0, y0), where x0 2 X0, y0 2 F(x0), to be a solution of this problem. We define and apply for this purpose the directional Dini derivative. Necessary conditions and sufficient conditions a pair (x0, y0) to be a w-minimizer, and similarly to be a i-minimizer are obtained. The role of the i-minimizers, which seems to be a new concept in set-valued optimization, is underlined. For the case of w-minimizers some comparison with existing results is done. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.
First order optimality condition for constrained set-valued optimization
A constrained optimization problem with set-valued data is considered. Different kind of solutions are defined for such a problem. We recall weak minimizer, efficient minimizer and proper minimizer. The latter are defined in a way that embrace also the case when the ordering cone is not pointed. Moreover we present the new concept of isolated minimizer for set-valued optimization. These notions are investigated and appear when establishing first-order necessary and sufficient optimality conditions derived in terms of a Dini type derivative for set-valued maps. The case of convex (along rays) data is considered when studying sufficient optimality conditions for weak minimizers. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.
The Impact of the CIMMYT Wheat Breeding Program on Mexican Wheat Producers and Consumers: An Economic Welfare Analysis
The increase in wheat production in Mexico’s Yaqui Valley from the breeding and development of semidwarf wheat varieties released by CIMMYT is quantified for the period 1990-2002, and the costs and benefits of the wheat research program are estimated and evaluated using a two-region model of the world wheat market.Public wheat breeding, benefit/cost analysis, agricultural research, wheat varieties, Crop Production/Industries, Research and Development/Tech Change/Emerging Technologies,
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